Theorem lcfrlem29

Description: Lemma for lcfr . (Contributed by NM, 9-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
	lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
	lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
	lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
	lcfrlem22.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
	lcfrlem24.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
	lcfrlem24.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
	lcfrlem24.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
	lcfrlem24.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
	lcfrlem24.j	⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
	lcfrlem24.ib	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
	lcfrlem24.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	lcfrlem25.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	lcfrlem28.jn	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
	lcfrlem29.i	⊢ 𝐹 = ( invr ‘ 𝑆 )
Assertion	lcfrlem29	⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ∈ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcfrlem17.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lcfrlem17.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lcfrlem17.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		lcfrlem17.p	⊢ + = ( +g ‘ 𝑈 )
6		lcfrlem17.z	⊢ 0 = ( 0g ‘ 𝑈 )
7		lcfrlem17.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8		lcfrlem17.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
9		lcfrlem17.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		lcfrlem17.x	⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
11		lcfrlem17.y	⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
12		lcfrlem17.ne	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
13		lcfrlem22.b	⊢ 𝐵 = ( ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ∩ ( ⊥ ‘ { ( 𝑋 + 𝑌 ) } ) )
14		lcfrlem24.t	⊢ · = ( ·𝑠 ‘ 𝑈 )
15		lcfrlem24.s	⊢ 𝑆 = ( Scalar ‘ 𝑈 )
16		lcfrlem24.q	⊢ 𝑄 = ( 0g ‘ 𝑆 )
17		lcfrlem24.r	⊢ 𝑅 = ( Base ‘ 𝑆 )
18		lcfrlem24.j	⊢ 𝐽 = ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↦ ( 𝑣 ∈ 𝑉 ↦ ( ℩ 𝑘 ∈ 𝑅 ∃ 𝑤 ∈ ( ⊥ ‘ { 𝑥 } ) 𝑣 = ( 𝑤 + ( 𝑘 · 𝑥 ) ) ) ) )
19		lcfrlem24.ib	⊢ ( 𝜑 → 𝐼 ∈ 𝐵 )
20		lcfrlem24.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
21		lcfrlem25.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
22		lcfrlem28.jn	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 )
23		lcfrlem29.i	⊢ 𝐹 = ( invr ‘ 𝑆 )
24	1 3 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
25	15	lmodring	⊢ ( 𝑈 ∈ LMod → 𝑆 ∈ Ring )
26	24 25	syl	⊢ ( 𝜑 → 𝑆 ∈ Ring )
27	1 3 9	dvhlvec	⊢ ( 𝜑 → 𝑈 ∈ LVec )
28	15	lvecdrng	⊢ ( 𝑈 ∈ LVec → 𝑆 ∈ DivRing )
29	27 28	syl	⊢ ( 𝜑 → 𝑆 ∈ DivRing )
30		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
31		eqid	⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 )
32		eqid	⊢ { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
33	1 2 3 4 5 14 15 17 6 30 20 21 31 32 18 9 11	lcfrlem10	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) )
34		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
35	1 2 3 4 5 6 7 8 9 10 11 12 13	lcfrlem22	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
36	34 8 24 35	lsatlssel	⊢ ( 𝜑 → 𝐵 ∈ ( LSubSp ‘ 𝑈 ) )
37	4 34	lssel	⊢ ( ( 𝐵 ∈ ( LSubSp ‘ 𝑈 ) ∧ 𝐼 ∈ 𝐵 ) → 𝐼 ∈ 𝑉 )
38	36 19 37	syl2anc	⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
39	15 17 4 30	lflcl	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑌 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
40	24 33 38 39	syl3anc	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 )
41	17 16 23	drnginvrcl	⊢ ( ( 𝑆 ∈ DivRing ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ≠ 𝑄 ) → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
42	29 40 22 41	syl3anc	⊢ ( 𝜑 → ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 )
43	1 2 3 4 5 14 15 17 6 30 20 21 31 32 18 9 10	lcfrlem10	⊢ ( 𝜑 → ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) )
44	15 17 4 30	lflcl	⊢ ( ( 𝑈 ∈ LMod ∧ ( 𝐽 ‘ 𝑋 ) ∈ ( LFnl ‘ 𝑈 ) ∧ 𝐼 ∈ 𝑉 ) → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 )
45	24 43 38 44	syl3anc	⊢ ( 𝜑 → ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 )
46		eqid	⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
47	17 46	ringcl	⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ∈ 𝑅 ∧ ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ∈ 𝑅 ) → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ∈ 𝑅 )
48	26 42 45 47	syl3anc	⊢ ( 𝜑 → ( ( 𝐹 ‘ ( ( 𝐽 ‘ 𝑌 ) ‘ 𝐼 ) ) ( .r ‘ 𝑆 ) ( ( 𝐽 ‘ 𝑋 ) ‘ 𝐼 ) ) ∈ 𝑅 )